This invention relates to a linear interpolating method for signals in a memory which is used for color correction of picture signals in a reproducing machine such as a color scanner, a color facsimile producer, or the like, in which color separation picture images are produced by photo-electric scanning, and the apparatus to carry out the method.
In conventional color photographic plate making, color correction is often made by photographic masking. However, this method has many defects, for example: limitations of color correction ability, necessity for many skilled engineers, unreliable results of the color separation, irregular quality of finish, complexity, and the like.
In order to overcome these defects, a color correction masking method by an electronic color separation machine such as a color scanner has been developed and is nowadays more popular. Most of the color scanners now used employ an analog computer system for the color correction calculations so as to increase the calculation speed.
This method, however, has also defects such as the difficulty of the introduction of many kinds of calculations because of the restriction of calculation ability, inevitable effects of temperature drift and noise, multiplicity of operational amplifiers and so forth as electric elements, inconvenience of operation due to many adjustments of potentiometers and switches, and high manufacturing cost.
If the analog computer system is simply replaced with a digital computer system, which has advantages such as a wide correction variable range and convenience of operation, the calculation speed for the color correction decreases very much, and the processing ability is reduced. Accordingly, this system is not practicable.
Recently, a direct scanner has been developed for plate making in printing, which performs color separation, color correction, conversion of scale of the reproduced image, and halftone processing at the same time so as to meet the requirement for high quality printing and rapid operation. In this case, however, there is the defect that supplementary masking or hand retouching after the color separation cannot be applied, as opposed to conventional color scanning which includes color separation, color correction, conversion of scale of the reproduction images, and halftone processing.
In general, an original color picture is scanned by a color scanner to obtain three (red, green, and blue) color separation signals. These three color separation signals are set to a color operation circuit, thereby finally obtaining recording signals for density of printing inks, such as cyan, magenta, yellow, and black.
In order to provide the most accurate possible color reproduction, a combination of the amounts of cyan, magenta, and yellow inks (the black ink, and so forth, are omitted for the sake of brevity of explanation) is necessarily determined corresponding to a combination of red, green, and blue color separation signals.
Consequently, for the purpose of color correction by selecting the combination of cyan, magenta, and yellow values corresponding to the combination of red, green, and blue values, the color-corrected combinations of cyan, magenta, and yellow values corresponding to each combination of red, green, and blue values are stored in a memory in advance, and then the color-corrected combination of cyan, magenta, and yellow values is read out by addressing the memory by the combination of red, green, and blue values corresponding thereto.
If each red, green, and blue range is divided into, for example, two hundred tone steps, altogether 200.sup.3 =8,000,000 combinations of cyan, magenta, and yellow values must be stored in the memory, which requires that the memory have a large capacity. This means high cost, and thus is not practicable.
Therefore, in order to reduce the storage capacity required for the memory, each color range of red, green, and blue is divided into, for example, sixteen tone steps, and then 16.sup.3 =4096 combinations of cyan, magenta, and yellow values are required. Thus the storage capacity requirement for the memory is reduced to a manageable level. On the other hand, the tone steps become too rough, and the lack of output consistency becomes conspicuous, so that printing quality suffers. Therefore, in this case, it is necessary to interpolate intermediate values properly between each two tone steps.
The present invention relates to an improved method of interpolation in the three-dimensional space defined in the memory by the three axes of red, green, and blue. In order that the method may be better understood, some explanation of prior art methods of interpolation will now be given.
Referring to FIG. 1, there is shown an example of interpolation of a function U of two variables, where the interval to be interpolated over is taken as unity.
The value U(x,y), i.e., U(x.sub.1 +x.sub.f, y.sub.i +y.sub.f) at a point P in an interpolation region ABCD will be found by a mathematical interpolating method, in which x.sub.i and y.sub.i are the intergral parts of x and y and x.sub.f and y.sub.f are the decimal parts.
For interpolation it is necessary that the function at the vertices A, B, C, and D should have known values U(x.sub.i,y.sub.i), U(x.sub.i +l,y.sub.i), U(x.sub.i +l,y.sub.i +l), and U(x.sub.i,y.sub.i +l). The interpolated value U(x,y) will be a function of X.sub.f, Y.sub.f, U(x.sub.i,Y.sub.i), U(x.sub.i +l,y.sub.i), U(x.sub.i +l,y.sub.l +l), and U(x.sub.i, y.sub.i +l). Further, for consistency, the interpolated value should be consistent with the known values of the original function at the corners of the unit region.
An interpolating method satisfying such a condition will be described. It is called linear interpolation because on the edges of the unit region it reduces to a simple linear interpolation function.
In order to find the value U(x,y) at the point P in the interpolation unit square ABCD, first drawn four perpendiculars from the point P to each side AB, BC, CD, and DA of the square. Designate the ends or feet of these perpendiculars by Q.sub.1, Q.sub.2, Q.sub.3, and Q.sub.4 respectively, as shown in FIG. 2, and add up the results obtained by multiplying each known value at the vertices A, B, C, and D by the area of each rectangle opposite to the vertex, thereby obtaining the following equation (I): ##EQU1##
The interpolating method according to the formula (I) satisfies the above boundary conditions at the corners of the unit square and reduces to linear interpolation along the edges of the unit square, and thus is mathematically reasonable. Further, this method may be applied to the three-dimensional case.
In FIG. 3 there is shown a unit cube interpolation unit having eight vertices with co-ordinates of (x.sub.i, y.sub.i z.sub.i), (x.sub.i +l,y.sub.i,z.sub.i), (x.sub.i,y.sub.i +l,z.sub.i), (x.sub.i,y.sub.i,z.sub.i +l), (x.sub.i +l,y.sub.i +l,z.sub.i), (x.sub.i +l,y.sub.i,z.sub.i +l), (x.sub.i,y.sub.i +l,z.sub.i +l), and (x.sub.i +l,y.sub.i +l,z.sub.i +l), and including a point P with co-ordinates (x.sub.i +x.sub.f, y.sub.i +y.sub.f, z.sub.i +z.sub.f) at which the value of U is to be interpolated. The cube is divided up into eight rectangular parallelepipeds by three planes which include the point P and are parallel to its faces. The value U(x,y,z) at the point P is found by adding up the values obtained by multiplying each known value at each of the vertices of the unit cube by the volume of each rectangular parallelepipedon which is positioned opposite to that vertex, thereby obtaining the following formula (II): ##EQU2##
Again, this method produces consistent results at the verticles of the unit cube. Further, along the edges of the unit cube it reduces to simple linear interpolation, and on the faces of the unit cube it reduces to the method of equation (I). It is further clear that the value obtained in the center of each face of the unit cube is the mean value of the known values at each vertex of that face, and the value obtained at the center of the unit cube is the mean value of the eight known values at the vertices of the cube. Accordingly, this method is seen to be mathematically reasonable.
However, this method has disadvantages. It requires eight products to be formed, each of four values, and addition thereof. Hence it is not always best for high speed calculation.
There is another disadvantage in this method Although from one unit cube to the next the interpolated values are continuous, their derivative is not. That is, the slope of the interpolated values is discontinuous from one unit cube to the next, i.e. the line of the interpolated values bends sharply as we pass over the boundary. Thus in practice a sharp step of color values will be apparent in the finished picture, and the cubic structure of the memory will show, to the detriment of quality. This effect can become quite serious. FIG. 4 shows a distribution of the interpolated values obtained according to the formula (I) which has a saddle form, which clearly shows the aforementioned problem. An even continuous line of interpolated values in the unit square A.sub.1 B.sub.1 C.sub.1 D.sub.1 is obtained, and also in the unit square A.sub.2 B.sub.2 C.sub.2 D.sub.2, but between these two squares, at their common border, the derivative of the interpolated values is discontinuous.